Subgroups — #27144693

Order 12416.

480 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260642	C₂	1	2	yes	maximal
2	#20260621	C₂	1	4	yes	maximal
3	#20260605	C₂	1	1	yes	maximal, characteristic
4	#20260601	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₉₇	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₉₇	C₂	1	2	yes	maximal
8	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
16	#15690717	C₂²	1	2	yes	—
17	#15690717	C₂²	1	4	yes	—
18	#15690717	C₂²	1	4	yes	—
19	#15690717	C₂²	1	4	yes	—
20	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₉₇	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₇	C₂³	1	4	yes	—
27	C₂²×Dic₉₇	C₂³	1	2	yes	—
28	C₂²×Dic₉₇	C₂³	1	2	yes	—
29	C₄⋊Dic₉₇	—	2	4	no	—
30	C₄⋊Dic₉₇	—	2	2	no	—
31	C₄⋊Dic₉₇	—	2	4	no	—
32	C₄⋊Dic₉₇	—	4	2	no	—
33	C₄⋊Dic₉₇	Q₈	1	4	yes	complemented
34	C₄×Dic₉₇	—	2	4	no	—
35	C₄×Dic₉₇	—	4	4	no	—
36	C₂×Q₈×C₉₇	—	2	2	no	—
37	C₂×Q₈×C₉₇	—	2	2	no	—
38	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₉₇	C₂×C₄	1	8	yes	—
45	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
46	C₄²×C₉₇	C₂×C₄	1	8	yes	—
47	C₄²×C₉₇	C₂×C₄	1	4	yes	—
48	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
49	Q₈×C₉₇	—	2	8	no	—
50	C₂×C₄×C₉₇	—	2	2	no	—
51	C₂×C₄×C₉₇	—	2	2	no	—
52	C₂×C₄×C₉₇	—	2	2	no	—
53	C₂×C₄×C₉₇	—	2	1	no	—
54	C₂×C₄×C₉₇	—	2	2	no	—
55	C₂×C₄×C₉₇	—	2	1	no	—
56	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₉₇	#41	1	2	yes	—
63	C₂×C₄×C₉₇	#41	1	2	yes	—
64	C₂×C₄×C₉₇	#41	1	2	yes	—
65	C₂×Dic₉₇	—	2	2	no	—
66	C₂×Dic₉₇	—	2	2	no	—
67	C₂×Dic₉₇	—	2	2	no	—
68	C₂×Dic₉₇	—	2	4	no	—
69	C₂×Dic₉₇	—	2	2	no	—
70	C₂×Dic₉₇	—	2	4	no	—
71	C₂×Dic₉₇	—	2	4	no	—
72	C₂×Dic₉₇	—	2	2	no	—
73	C₂×Dic₉₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
77	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
78	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₉₇	—	2	4	no	—
82	C₄×C₉₇	—	2	4	no	—
83	C₄×C₉₇	—	2	2	no	—
84	C₄×C₉₇	#117	1	4	yes	—
85	C₄×C₉₇	C₄×Q₈	1	4	yes	—
86	Dic₉₇	—	2	4	no	—
87	Dic₉₇	—	4	2	no	—
88	Dic₉₇	—	4	4	no	—
89	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
90	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₉₇	#520	1	1	yes	characteristic
92	C₂×C₉₇	#554	1	1	yes	characteristic
93	C₂×C₉₇	#556	1	1	yes	characteristic
94	C₂×C₉₇	#557	1	1	yes	characteristic
95	C₂×C₉₇	#559	1	1	yes	characteristic
96	#2355	—	97	1	no	maximal, Sylow
97	C₉₇	#2355	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
99	#391	—	97	2	no	—
100	#387	—	97	4	no	—
101	#384	—	97	1	no	—
102	#382	—	97	2	no	—
103	C₄×C₄⋊C₄	—	97	2	no	—
104	C₄×C₄⋊C₄	—	97	1	no	—
105	C₄×C₄⋊C₄	—	97	2	no	—
106	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
107	C₄×Q₈	Dic₉₇	1	8	yes	complemented
108	C₂×C₄⋊C₄	—	97	1	no	—
109	C₂×C₄⋊C₄	—	97	2	no	—
110	C₂×C₄⋊C₄	—	97	1	no	—
111	C₂×C₄⋊C₄	—	97	2	no	—
112	C₂×C₄⋊C₄	—	97	2	no	—
113	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
115	C₂×C₄²	—	97	4	no	—
116	C₂×C₄²	—	97	2	no	—
117	C₂×C₄²	C₂×D₉₇	1	2	yes	—
118	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
119	#95	—	97	4	no	—
120	#95	—	97	2	no	—
121	#95	—	97	4	no	—
122	#95	—	97	4	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	C₂×Dic₉₇	1	4	yes	—
126	C₂²×C₄	—	97	2	no	—
127	C₂²×C₄	—	97	2	no	—
128	C₂²×C₄	—	97	4	no	—
129	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
130	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
131	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
134	C₄⋊C₄	—	97	4	no	—
135	C₄⋊C₄	—	388	2	no	—
136	C₄⋊C₄	—	194	4	no	—
137	C₄⋊C₄	—	194	2	no	—
138	C₄⋊C₄	—	194	4	no	—
139	C₄⋊C₄	C₂×Dic₉₇	1	8	yes	—
140	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
141	C₄²	—	388	4	no	—
142	C₄²	—	194	4	no	—
143	C₄²	C₂×Dic₉₇	1	8	yes	—
144	C₄²	C₂×Dic₉₇	1	4	yes	—
145	C₂³	C₂³×D₉₇	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	97	4	no	—
154	C₂×C₄	—	194	2	no	—
155	C₂×C₄	—	194	4	no	—
156	C₂×C₄	—	194	2	no	—
157	C₂×C₄	—	194	4	no	—
158	C₂×C₄	—	194	4	no	—
159	C₂×C₄	—	194	2	no	—
160	C₂×C₄	—	194	2	no	—
161	C₂×C₄	—	194	2	no	—
162	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
163	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
164	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
165	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
166	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
167	C₂×C₄	Q₈×D₉₇	1	2	yes	—
168	C₂×C₄	#13095511	1	2	yes	—
169	C₂×C₄	#13095511	1	2	yes	—
170	C₂×C₄	#13095512	1	2	yes	—
171	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
173	C₂²	C₂×#13095511	1	1	yes	characteristic
174	C₂²	C₂×#13095511	1	1	yes	characteristic
175	C₂²	C₂×#13095512	1	1	yes	characteristic
176	C₂²	#15690856	1	1	yes	characteristic
177	C₂²	#15690857	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	388	2	no	—
182	C₄	—	388	4	no	—
183	C₄	—	194	4	no	—
184	C₄	#15690769	1	4	yes	—
185	C₄	Q₈×Dic₉₇	1	4	yes	—
186	C₂	C₂×#15690769	1	1	yes	characteristic
187	C₂	C₂×Q₈×Dic₉₇	1	1	yes	characteristic
188	C₂	#20261218	1	1	yes	characteristic
189	C₂	#20261445	1	1	yes	characteristic
190	C₂	#20261458	1	1	yes	characteristic
191	C₂	#20261463	1	1	yes	characteristic
192	C₂	#20261470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₉₇	C₂
socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#2355
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
21. C₄×Q₈×C₉₇	C₄	semidirect
33. C₄⋊Dic₉₇	Q₈	semidirect
97. C₉₇	#2355	semidirect
107. C₄×Q₈	Dic₉₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20260642	2
#20260621	2
#20260605	2
#20260601	2
C₄⋊C₄×Dic₉₇	2
C₄×C₄⋊Dic₉₇	2
C₂×C₄×Q₈×C₉₇	2
#2355	97